alignment tests for low cmb multipoles

Alignment tests for low CMB multipoles

L. Raul Abramo*Instituto de Fısica, Universidade de Sao Paulo CP 66318, CEP 05315-970 Sao Paulo, Brazil

Armando Bernui,† Ivan S. Ferreira,‡ Thyrso Villela,x and Carlos Alexandre Wuenschek

Divisao de Astrofısica, Instituto Nacional de Pesquisas Espaciais Av. dos Astronautas, 1.758, CEP 12227-010,Sao Jose dos Campos, Brazil

(Received 10 May 2006; published 7 September 2006)

We investigate the large scale anomalies in the angular distribution of the cosmic microwavebackground radiation as measured by WMAP using several tests. These tests, based on the multipolevector expansion, measure correlations between the phases of the multipoles as expressed by thedirections of the multipole vectors and their associated normal planes. We have computed the probabilitydistribution functions for 46 such tests, for the multipoles ‘ � 2� 5. We confirm earlier findings thatpoint to a high level of alignment between ‘ � 2 (quadrupole) and ‘ � 3 (octopole), but with our tests wedo not find significant planarity in the octopole. In addition, we have found other possible anomalies in thealignment between the octopole and the ‘ � 4 (hexadecupole) components, as well as in the planarity of‘ � 4 and ‘ � 5. We introduce the notion of a global anomaly statistic to estimate the relevance of thelow-multipoles tests of non-Gaussianity. We show that, as a result of these tests, the CMB maps which aremost widely used for cosmological analysis lie within the�10% of randomly generated maps with lowestglobal anomaly statistics.

DOI: 10.1103/PhysRevD.74.063506 PACS numbers: 98.80.Es, 98.65.Dx, 98.70.Vc

I. INTRODUCTION

The cosmic microwave background (CMB) anisotropieshave been measured with exquisite accuracy by theWilkinson Microwave Anisotropies Probe (WMAP), andthe impact on cosmology has been profound [1– 4].However, as the �CDM cosmological model becomesstandard lore and the parameter space becomes narrower,the focus naturally drifts to the apparent anomalies. Amongsources of concern that have survived the WMAP 3-yeardata are the lack of large-angle correlations [1,2,4], whichis mainly due to the low value of the cosmic quadrupole[4–8], and the alignment between the quadrupole (‘ � 2)and octopole (‘ � 3) [4,5,9–15].

The combined statistics of the low quadrupole and of thequadrupole-octopole alignment, which were already in theCOBE data [16,17], can only be matched by 0.01% ofGaussian random maps—see, for instance, [5]. However,caution should be taken in the interpretation of that resultas an outright indication of violations of statistical isotropy,since this tiny probability was in fact constructed a poste-riori, by multiplying the probabilities of two statisticaltests which were previously thought to be unrelated.

Several recent works have reported anomalies in thedata: the low-order multipole values [1,2,4,6,15,18–20];the alignment of some low-order multipoles [5,15,21–23];an unexpected asymmetric distribution on the sky of the

large-scale power of CMB data [14,24–27]; indications fora preferred direction of maximum asymmetry [9,25,27–30]; as well as apparent non-Gaussian features detected viathe wavelet method or other analyses [31–36]. Theseanomalies have motivated many explanations, such ascompact topologies [37,38], a broken or suppressed spec-trum at large scales [39–44], oscillations superimposed onthe primordial spectrum of density fluctuations [45–47],anisotropic cosmological models [48–50] and possibleextended foregrounds that could be affecting the CMB[10,51–54].

In this article we examine the multipoles ‘ � 2� 5 andsearch for anomalies in their phase correlations. We mea-sure these correlations through alignments between themultipole vectors or through their associated normal planes[9]. We also look for evidence of planarity (or self-alignments) in each individual multipole, and for evidenceof alignments between the multipole and normal vectorswith some specific directions in the sky, such as the dipoleaxis, the ecliptic axis and the Galactic poles axis. In totalwe have considered 38 tests of alignments between multi-poles, plus 8 tests of alignments of the multipoles with apriori directions. We have computed the probability distri-bution functions (PDFs) for those tests using 300 000 mockmaps.

For the statistical analyses performed here, we need thea‘m’s (‘ � 2� 5) of each CMB map under investigation.The corresponding a‘m’s were extracted, after applying theKp2 WMAP mask (to minimize foreground contamina-tions from the beginning), through the HEALPix routines[55]. Our statistical tools were then used to analyze theWMAP 1-year and 3-year data Internal Linear Com-

*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected] address: [email protected] address: [email protected]

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bination maps [1,4] (henceforth ILC), the co-added1-year and 3-year WMAP data, as well as the cleanedCMB maps of Tegmark et al. [13,15] based on 1-yearWMAP data (henceforth TOH).

This paper is organized as follows. In Sec. II we sum-marize the multipole vector formalism and the severaldifferent statistics that can be used to test for alignmentsand phase correlations within a given multipole. In Sec. IIIwe briefly describe the CMB maps used. Section IVpresents our statistical tests and the results of the PDFcomputations for those tests. We also analyze the salientfeatures of the CMB maps and discuss which tests can beconsidered anomalous, and of those, which are robust andwhich are most sensitive to noise. The conclusions arepresented in Sec. V.

II. MULTIPOLE VECTORS, NORMAL VECTORSAND STATISTICS OF PHASE CORRELATIONS

Multipole vectors were introduced in CMB data analysisby Copi et al. [9], and Katz and Weeks [11,12] found anelegant algebraic method to compute the multipole vectorsgiven the spherical harmonic components a‘m—see also[56] for an alternative algebraic method and [5,10,14] forpurely numerical methods. The multipole vectors are es-sentially eigenvectors—i.e., they are solutions of a set ofpolynomial equations whose parameters are the multipolecomponents a‘m.

The idea of the multipole vector representation goesback to J. C. Maxwell in the XIXth century: the multipoledecomposition of a field f��;�� on S2 implies that for eachmultipole ‘ there are ‘ eigenvectors of norm unity, n�‘;p�.Since there are only 2‘ phases for each multipole, thespherical harmonic representation and the multipole vectorrepresentation have the same number of degrees of free-dom in each individual multipole:

�T‘��;’�T

�X‘

m��‘

a‘mY‘m��; ’�

� D‘

Y‘p�1

n�‘;p� � n��;�� Z‘�1��;’�; (1)

where Z‘�1 just subtracts the residual ‘0 < ‘ total angularmomentum parts of the product expansion, and is irrelevantto our analysis—see [11] for an enhanced discussion of themultipole vector expansion.

It can be seen from the product expansion above that,whenever using the multipole vectors to test for align-ments, it is irrelevant what the amplitudes of the multipolesare—just their phases matter. This is the main feature ofthe tests based on the multipole vectors that sets them apartfrom other tests of non-Gaussianity.

Notice that, contrary to the C‘’s, which are alwayspositive-definite, the D‘’s of Eq. (1) can be either negativeor positive. Because of the product expansion in the right-

hand-side of Eq. (1), switching the sign of D‘ is equivalentto switching the signs of an odd number of multipolevectors, and switching the signs of an even number ofmultipole vectors leaves the sign of D‘ invariant.Therefore, the product expansion in Eq. (1) has a signdegeneracy in the amplitudesD‘ as well as in the multipolevectors n�‘;p�.

This ‘‘reflection symmetry’’ n�‘;p� $ �n�‘;p� impliesthat the multipole vectors define only directions [11],hence they are ‘‘vectors without arrowheads’’ living onthe half-sphere with antipodal points identified, or S2=Z2.This space is also known in the literature as the realprojective space RP2, and is useful in the characterizationof nematic liquid crystals, where the orientation of themolecules is an order parameter—but it makes no differ-ence where heads and tails are [57].

It is extremely useful to represent these directions asvectors, but for that we will need to cope with the degen-eracy in representing these directions. For each multipoleorder ‘ there is a 2‘�1-fold degeneracy in the signs (ororientations) of the multipole vectors—corresponding tothe ‘ signs of the multipole vectors that can be switchedarbitrarily, divided by two to account for an irrelevantoverall sign which is determined by D‘. We can breakthis degeneracy by always working in one particular hemi-sphere, and any such choice will automatically determinethe signs (orientations) of all multipole vectors—as well asthe sign of the D‘’s. However, we should be aware that indoing so we are necessarily picking one of the 2‘�1 pos-sible sign conventions in the product expansion of Eq. (1).As we will discuss below, this is not a problem as long aswe use invariant tools which are not sensitive to the signs ofeach multipole vector.

Starting with the ‘ multipole vectors one can also con-struct ‘�‘� 1�=2 � � normal vectors—or normalplanes—defined as:

~w ‘;q � n‘;p ^ n‘;p0; �p � p0; q � 1 . . .��: (2)

By construction, because the multipole vectors defineonly directions, the normal vectors also possess reflectionsymmetry, ~w�‘;q� $ � ~w�‘;q�. But the normal vectors neednot be (and generally are not) of norm unity, so instead ofliving in S2=Z2 the normal vectors belong to the spaceR3=Z2—which is isomorphic to SO(3), see [57]. We canstill break the degeneracies imposed by reflection symme-try by defining all normal vectors so they lie in the samehemisphere as the normal vectors. However, just as before,we should be aware that in doing so we are choosing one ofmany possible representations for the normal vectors.

To summarize, for ‘ � 2 there are 2 multipole vectors,n�2;1� and n�2;2�, and only one normal vector, ~w�2;1� �n�2;1� � n�2;2�; for ‘ � 3 there are 3 multipole vectors and3 normal vectors; and so forth. These constitute the basisfor the statistical tests defined below.

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A. Properties of the tests under reflection symmetry

We will define below, in Sec. II C, a series of tests whichare manifestly invariant under the reflection symmetry n$�n which characterizes RP2 (where the multipole vectorslive) as well as R3=Z2 (where the associated normal vec-tors live.)

Our motivation for this remark is that if a given test is notinvariant then its validity and usefulness is questionable. Inparticular, one should be careful not to employ tests whichdepend on the choice of hemisphere to represent the vec-tors. This seems to be the case of some of the tests that havebeen used to estimate the ‘‘planarity’’ of the CMB maps, ifthese tests make use, in one way or another, of the notion of‘‘average vectors’’. The reason there is no such thing as an‘‘average multipole vector’’ or an ‘‘average normal vector’’is simple: the ‘‘vector sum’’ operation does not yield asingly valued result due to the reflection symmetry. In fact,the result of ‘‘summing’’ two vectors of R3=Z2 would be adegenerated pair of directions:

� ~v1� � ~v2� �

� ~v1 ~v2

~v1 � ~v2: (3)

In general, by summing k vectors of R3=Z2 one obtains2k�1 vectors, corresponding to the 2k possible permuta-tions of the signs of each vector, divided by two toaccount for the symmetry ~v$ � ~v of the resulting vector.This means, in particular, that the ‘‘average multipolevector’’ is in fact an object 2‘�1-times degenerated,and that the ‘‘average normal vector’’ is an object2‘�‘�1�=2�1-times degenerated.

Obviously, by fixing a hemisphere to represent all vec-tors one breaks this degeneracy, but this just hides the plainfact that by doing so one is simply choosing (rather arbi-trarily) one of many possible representations, and one ofmany possible answers for the sums of those vectors.Consequently, unless these degeneracies are properly takeninto account (by, e.g., symmetrizing over all possibilities orordering the results by norm), any test which is derivedfrom the notion of summing multipole or normal vectors isflawed.

B. Global estimates of non-Gaussianity

We will investigate large-scale correlations in the CMBmaps within single multipoles and between different multi-poles by measuring the alignments between either themultipole vector themselves, or between their associatednormal planes. To be sure, there is no upper limit to thenumber of tests we can devise to search for non-Gaussianities, and in testing any fixed sample such as theCMB one should bear in mind that there are always somestatistical tools which will yield a positive detection givensome arbitrary criteria.

In practical terms this means that if we perform a largenumber N of independent tests on a fixed sample, then weshould treat the results of these tests themselves as random

numbers. Therefore, when performing many tests on a mapand searching for clues of non-Gaussianity one must al-ways look at the complete set of results for those tests andat the total probability (or some global statistic) that themap is a realization of a Gaussian random process. If thisglobal statistic turns out to be very small compared to thetypical values for Gaussian maps, then one can look for theparticular test (or tests) that is likely responsible for thatanomaly. If, however, a certain test turns out to be ‘‘suspi-cious’’ but the global statistic is not anomalously small,then we cannot rule out the possibility that the result forthat particular test was just a fluke.

Assuming that random processes are indeed behind themechanism that generated the sample, we can define aglobal anomaly statistic in the following sense. Supposewe have N statistical tests such that each test Ti (i �1 . . .N ) is a random number in the interval 0 � Ti � 1,with normalized probability distribution functions Pi�Ti�.Given a sample (a map M) with Ti � TMi , the probabilitythat a random sample has a value of Ti higher than TMi is

Pi �TMi � �Z 1

TMi

dTPi�T�; (4)

and the probability that a random sample has a value of Tilower than TMi is

Pi��TMi � �

Z TMi

0dTPi�T� � 1� Pi �T

Mi �: (5)

Evidently, for the median value �Ti we have Pi � �Ti� �Pi�� Ti� � 1=2. We will define the global median statisticof the map M, given the N tests, to be

LN �M� �YNi�1

2Pi �TMi � � 2Pi��TMi �

� 4NYNi�1

Pi �TMi ��1� Pi �T

Mi ��; (6)

where the factors of 2 have been inserted for normalizationpurposes, in order to make LN � �M� � 1 for a map �Mwhose tests are all exactly equal to their median values.

This global anomaly statistic estimates the total proba-bility that the map M does not have too low and too highvalues of the tests Ti. Evidently, any deviation of the testsfrom the medians will decrease LN . So, the global anom-aly statistic is an unbiased estimator of the degree ofanomalies of a given map, in the sense that both too highand too low correlations would be considered anomalous.

Of course, Eq. (6) is itself an arbitrary definition, and wemight as well have used the expectation values instead ofthe medians to define the global statistic. Indeed, one couldswitch the roles of the medians with the expectation valuesin the procedure above and still our results would be verysimilar. Ideally, we would have computed the Likelihoodfunction, but it turns out that for the numerical strategy we

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pursued in this particular work, it would have been im-practical to do so.

It should be stressed that the global statistic defined inthe sense above should not be interpreted as the probabilitythat a particular map was generated by a Gaussian mecha-nism—it is merely an estimator of how much that particu-lar map deviates from a typical one, given the N statisticaltests of non-Gaussianity. In Sec. II D we construct suchtests, and compute their distributions assuming randomphases.

C. Statistical tests of isotropy

We now define the statistical tests which will be em-ployed in our analysis of CMB data. We have ensured thatall tests are invariant under reflection symmetry, so it

makes no difference which hemisphere one chooses torepresent the multipole and normal vectors.

The tests have been normalized so that they always fallin the interval 0 � Ti � 1. We have generated 3� 105simulated (mock) maps by taking Gaussianly distributedrandom spherical harmonic components. The resultingPDF’s for the tests, assuming Gaussianity, are shown inFigs. 1–4.

1. S statistic

The S statistic is a widely used tool [9,11,14,27], and itmeasures the alignment between normal planes of differentmultipoles. It is defined as

S‘‘0 �1

��0X�q�1

X�0q0�1

j ~w�‘;q� � ~w�‘0;q0�j; ‘ � ‘0; (7)

where, as defined above, � � ‘�‘� 1�=2. We can also useS in just one multipole, in which case the normalization is a

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FIG. 1 (color online). Normalized PDF’s for the S statisticfound by simulating 3� 105 mock maps, binned in intervalsof 0.01. From left to right, top to bottom: S22, S23, S24, S25, S33,S34, S35, S44, S45 and S55. In all panels, the horizontal axiscorrespond to the value of each individual test, and the verticalaxis to its normalized PDF.

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FIG. 2 (color online). Normalized PDF’s for the D statistic.From left to right, top to bottom: D23, D24, D25, D33, D34, D35,D44, D45 and D55.


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bit different:

S‘‘ �2

�� 1�

X�q;q0>q

j ~w�‘;q� � ~w�‘;q0�j: (8)

The statistic S‘‘ measures the ‘‘self-alignment’’ of thenormal vectors, and is related to the planarity tests [5,14].

2. D statistic

This is analogous to the S statistic, but the D testdisregards the norm of the normal vectors [9,11,14,27]:

D‘‘0 �1

��0X�q�1

X�0q0�1

jw�‘;q� � w�‘0;q0�j; ‘ � ‘0: (9)

We can also use theD statistic within a single multipole, aswas done for S. However, this test only gives nontrivialinformation for ‘ � 3. With the proper normalization wehave

D‘‘ �2

�� 1�

X�q;q0>q

jw�‘;q� � w�‘;q0�j; ‘ � 3: (10)

3. R statistic

A similar tool is the R statistic, which measures align-ments in essentially the same way as the S statistic, but ituses the multipole vectors instead of the normal vectors:

R‘‘0 �1

‘‘0X‘p�1

X‘0p0�1

jn�‘;p� � n�‘0;p0�j; ‘ � ‘0: (11)

Within a single multipole, the R statistic is suitably definedas

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FIG. 3 (color online). Normalized PDF’s for the R statistic.From left to right, top to bottom: R22, R23, R24, R25, R33, R34,R35, R44, R45 and R55.

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FIG. 4 (color online). Normalized PDF’s for the N and W statistics—alignments of the multipole (N) and normal (W) vectorswith a particular direction in the sky. Top line, from left to right: N2, N3, N4, N5. Bottom line, from left to right: W2, W3, W4,W5.


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R‘‘ �2

‘�‘� 1�

X‘p;p0>p

jn�‘;p� � n�‘;p0�j: (12)

4. B statistic

We can also test if the multipole vectors align with thenormal vectors. Hence we define the B statistic:

B‘‘0 �1

�‘0X�q�1

X‘0p0�1

j ~w�‘;q� � n�‘0;p0�j; ‘ � ‘0: (13)

Within a single multipole, the B statistic only gives non-trivial information for ‘ � 3, and we have

B‘‘ �1

��‘� 2�

X�q

X‘p

j ~w�‘;q� � n�‘;p�j; ‘ � 3: (14)

Notice that, as opposed to the S, D and R statistics, the Bstatistic for ‘ � ‘0 is not symmetric, B‘‘0 � B‘0‘. Forsimplicity, in the present approach we have only consid-ered the cases B‘‘0 where ‘ � ‘0.

5. N statistic

We can test if the multipole vectors align in a particulardirection Z by using the N statistic:

N‘ �1

‘

X‘p�1

jn�‘;p� � Zj: (15)

6. W statistic

We can also test if the normal vectors align in a particu-lar direction Z by using the W statistic:

W‘ �1

�

X‘q�1

j ~w�‘;q� � Zj: (16)

Note that the test S�4;4� of Ref. [14] is a combination of W2

and W3, namely, S�4;4� � �W2 3W3�=4.

D. Likelihoods

With the tests defined above we can now compute globalstatistic as in Eq. (6). However, we have decided not toinclude the alignment testsN andW in the analysis, as theytest correlations with an a priori direction, which we findrather arbitrary compared to the other tests. Nevertheless, itshould be noted that there are significant correlations be-tween the quadrupole and the octopole with both theecliptic plane and the direction of the cosmic dipole.Although these correlations appear to be too strong ortoo weak only at >95% C.L. as measured by our tests N‘and W�, when combined in the statistic S�4;4� of Copi et al.[27], we obtain a result which is >99:5% C.L. These

correlations have been treated in much greater detail inRefs. [9,14,21,27].

Using the 38 tests S, D, R and B above, we can define aglobal anomaly statistic, L38�S;D; R; B�, as in Eq. (6). Wehave computed L38 for 25 000 random maps, and we showa normalized histogram for log10L38 in Fig. 5 (left panel.)

Since we have 38 tests but only 28 independent randomdegrees of freedom in the ‘ � 2� 5 multipoles, we havefound useful to define global anomaly statistic using asubset of the complete set of tests. We have thus definedthe statistic L29�S;D; R� using the tests S, D and R, andL29�S; R;B�, using the S, R and B tests. Their normalizedhistograms are also shown in Fig. 5 (center and rightpanels.)

III. CMB MAPS

In this work, we use five WMAP CMB maps (threederived from the 1-year data [1] and two derived fromthe 3-year data [58]): the 1-year and 3-year co-addedmaps [59]; the 1-year and 3-year ILC maps [3,58]; andthe TOH map [15].

The co-added WMAP map results from the combinationof the eight differential assemblies (DA) [59] in the Q-, V-,and W-bands, listed above, using the following inverse-variance noise weights method. Thus, for any co-addedmap, the temperature of pixel n is given by

T�n� �

P8i�1 Ti�n�=�

2i �n�P8

i�1 1=�2i �n�

; (17)

where Ti�n� is the sky map for the DA iwith the foregroundgalactic signal subtracted, and where

�2i �n� � �2

0;i=Nobsi�n� (18)

is the pixel-noise per observation for DA i.The eight valuesof �2

0;i are the pixel-noise for DA i, and can be found foreach set of released maps in [60].

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FIG. 5 (color online). Normalized histograms oflog10L38�S;D; R; B� (left panel), log10L29�S;D; R� (center) andlog10L29�S; R; B� (right), obtained by simulating 25 000 mockmaps.


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The WMAP 1-year Internal Linear Combination map(ILC-1 yr) [3] has been built mostly to convey a visualinformation of CMB anisotropies. It is composed by thefive WMAP temperature intensity maps, through aweighted linear combination, to minimize Galactic fore-ground contamination in 12 regions of the sky, eleven ofwhich lie within the Galactic plane and one which liesoutside it. The ILC-1 yr map is not reliable for quantitativeCMB analysis, but in this work we use it only for thecompleteness of our analysis. The 3-year WMAP ILCmap (ILC-3 yr) [58], however, brings some improvementsover the ILC-1 yr, mainly by dealing with the regionsselected for Galactic foreground estimates, which make ita reliable estimator of the CMB signal for large angularscales (> 10�) and, therefore, suitable for our analysissince we are interested in ‘ � 2� 5.

The 1- and 3-yr ILC maps are described, respectively, in[3,58], but it is worth mentioning here the differencesbetween them. The 1-yr and 3-yr ILC maps are bothweighted combinations of the maps, band-averaged withineach of the five WMAP frequency bands. They correctedthe bias due to the foreground removal method by dividingthe sky into 12 regions, minimizing the variance of thelinear combination of the 5 frequencies for each region andthen weighting the final combined map according to theregion. There was an additional bias correction, in the 3-yrILC map, that was done after using 100 Monte Carlosimulations with a variable spectrum Galaxy model asinput [58]. The WMAP team claims the residual error inthe 3-year ILC map is less than 5 �K in angular scalesgreater than 10�, being a reliable estimate of the CMBsignal, with negligible instrument noise, over the full sky.

The TOH map [15] was constructed with no assumptionabout the CMB power spectrum, foregrounds or noiseproperties. The only consideration was the Planckian na-ture of the CMB temperature spectrum. This map is formedby the combination of the five WMAP bands consideringweights that depend both on angular scale and distance tothe Galactic plane. The cleaning process is done in multi-pole space alm, weighted also by the beam function of eachchannel (W-band has four channels, Q- and V-band havetwo channels each, and K- and Ka-band have one channeleach). They obtain weighting coefficients similar to thoseused by the WMAP team on large scales.

IV. STATISTICS OF LARGE-ANGLEANISOTROPIES

Given a CMB map, the harmonic components a‘m canbe extracted using HEALPix [55], and the multipole vec-tors and their statistics can be easily computed. In thisanalysis we will use the five maps described above, plusthe TOH map cleaned with the mask ‘‘M6’’ given in[13,15]. For all maps we have used the Kp2 mask basedon 3-year WMAP data [3,4], then we remove their residual

monopole and dipole components. It should also be notedthat the relativistic Doppler correction to the quadrupole isan important factor that must be subtracted from the maps,since it corresponds to a nonprimary source of the quad-rupole [14,27].

In Table I we present an abridged version of the tests thatwere defined in Sec. II C. For an easier interpretation of theresults, we show the probabilities P �Ti� that a randommap would have higher values for the test Ti than the valuefor that test in the actual CMB maps. So, for instance, forthe test S22 and for the co-added 1-yr map we quote thevalue P �S22� � 0:294. This means that the probabilitythat a random map has a value of S22 which is higherthan the value obtained for the map co-added 1-yr is29.4%. Therefore, in Table I, any values which are tooclose to either zero or one should be viewed with suspicion.

It can be immediately seen from Table I that thequadrupole-octopole alignment (revealed by S23, D23 andB23) is robust in all maps, as has been noted by[5,9,13,14,20,21,27]. Our results confirm that the proba-bility that a random map has higher quadrupole-octopolealignment, as measured by S23, is approximately 1–2%.

Table I also reveals (through the self-alignments S33,D33, R33 and B33) that, at least for our set of invariant tests,there is no evidence that the octopole is significantly‘‘planar,’’ in the sense defined by these tests.

There are other anomalies for ‘ � 3 as well: we find thatthe octopole and hexadecupole (‘ � 4) are misaligned to avery significant degree, with a probability in the range 1%–4% for the test S34 and 5%–20% for the tests D34 and R34.The triandabipole (‘ � 5) also seems substantially mis-aligned with the octopole ‘ � 3, with a probability in therange 4%–8% for the test S35. These results, plus the factthat the octopole is significantly aligned with the dipoleaxis, give further support to the conjecture known as the‘‘axis of evil’’ [21].

Another apparent anomaly that is revealed by Table I isthe self-alignment of the multipoles ‘ � 4 and ‘ � 5,indicated by the values of P for S44, D55, R44 and R55.

Not shown in Table I are the results for the tests N andW, using as a priori directions to be tested against thedipole axis, the ecliptic plane and the galactic plane. Theresults for these 24 tests indicate that there are significantalignments of the quadrupole and octopole with the direc-tion defined by the cosmic dipole [5,9,13,14,20,21,27]:WDipole

2 � 0:10� 0:01 and WDipole3 � 0:03–0:05 depending

on the choice of map. We also confirm the results of Copiet al. [9,14,20,27], who found strong correlations of thequadrupole and octopole with the ecliptic plane: we getWEcliptic

2 � 0:83–0:98 and WDipole3 � 0:93–0:98 depending

on the choice of map. If combined into a single test, thetests W2 and W3 give rise to very significant correlations,both with the cosmic dipole and with the ecliptic plane. Forall other multipoles and directions we have found nosignificant alignments using our tests.


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TABLE II. Total likelihoods L38�S;D; R; B�, L38�S;D; R� and L29�S; R; B�. The normalized histograms for the global anomalystatistics found by simulating 25 000 maps are shown in Fig. 5.

Global Statistics Co-added 1-yr Co-added 3-yr ILC 1-yr ILC 3-yr TOHMask6 TOHKp2

L38�S;D; R; B� 3:2� 10�13 2:0� 10�14 7:7� 10�18 3:6� 10�14 3:7� 10�16 4:8� 10�12

L29�S;D; R� 1:7� 10�11 4:3� 10�13 5:2� 10�15 5:1� 10�13 3:0� 10�13 1:3� 10�10

L29�S; R; B� 5:1� 10�10 4:3� 10�11 3:0� 10�12 8:8� 10�11 3:9� 10�12 3:6� 10�9

TABLE I. Probabilities P for the tests S,D, R and B. These are the probabilites that a randommap would have values for the tests S, D, R and B which are higher than the map’s values. Theprobability that a random map has a value lower than the map’s is simply P� � 1� P .

Statistic Co-added 1-yr Co-added 3-yr ILC 1-yr ILC 3-yr TOHMask6 TOHKp2

S22 0.294 0.242 0.242 0.437 0.437 0.294S23 0.011 0.011 0.007 0.021 0.006 0.017S24 0.519 0.637 0.519 0.803 0.774 0.519S25 0.922 0.903 0.937 0.903 0.864 0.913S33 0.457 0.530 0.457 0.549 0.270 0.586S34 0.965 0.973 0.978 0.978 0.990 0.965S35 0.957 0.943 0.957 0.921 0.921 0.921S44 0.960 0.960 0.975 0.985 0.905 0.960S45 0.750 0.750 0.750 0.750 0.750 0.750S55 0.714 0.714 0.714 0.714 0.664 0.689

D23 0.034 0.046 0.031 0.056 0.019 0.051D24 0.771 0.884 0.771 0.922 0.937 0.085D25 0.854 0.786 0.907 0.744 0.550 0.786D33 0.268 0.333 0.276 0.351 0.187 0.361D34 0.802 0.888 0.916 0.916 0.965 0.850D35 0.711 0.711 0.797 0.607 0.375 0.607D44 0.913 0.913 0.913 0.913 0.623 0.913D45 0.342 0.342 0.342 0.342 0.786 0.499D55 0.970 0.970 0.989 0.970 0.970 0.970

R22 0.828 0.931 0.880 0.647 0.647 0.811R23 0.071 0.030 0.091 0.091 0.176 0.115R24 0.576 0.767 0.710 0.896 0.817 0.576R25 0.823 0.759 0.914 0.759 0.598 0.759R33 0.424 0.506 0.424 0.533 0.294 0.533R34 0.706 0.902 0.937 0.937 0.986 0.902R35 0.834 0.834 0.899 0.834 0.834 0.834R44 0.892 0.939 0.968 0.968 0.485 0.939R45 0.338 0.198 0.198 0.198 0.670 0.338R55 0.977 0.977 0.977 0.977 0.945 0.945

B23 0.940 0.920 0.920 0.920 0.968 0.883B24 0.139 0.192 0.065 0.192 0.033 0.117B25 0.344 0.256 0.392 0.256 0.806 0.298B33 0.696 0.582 0.644 0.540 0.749 0.554B34 0.651 0.426 0.344 0.262 0.189 0.506B35 0.591 0.414 0.506 0.414 0.784 0.414B44 0.079 0.116 0.201 0.249 0.300 0.116B45 0.336 0.336 0.497 0.497 0.497 0.336B55 0.444 0.444 0.338 0.338 0.136 0.233


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We can estimate the global statistics for the CMB mapsof Table I. This is shown in Table II for the statisticsL38�S;D; R; B�, L29�S;D; R� and L29�S; R; B�. Comparingthe values of Table II with the normalized histograms ofFig. 5, we can see that, apart from the TOH map with theKp2 mask applied, which lies within the �30% of randommaps with lowest global anomaly statistics, all remainingmaps fall within the �10% of random maps with lowestglobal anomaly statistics. The global statistics are particu-larly small for the 1-year ILC map, which, as discussed, isprobably contaminated by residual foregrounds.

V. CONCLUSIONS

We have applied 38 tests of non-Gaussianity, as well as 8tests of alignments with 3 distinct preferred directions, onthe most widely used CMB maps based on 1-year and 3-year WMAP data. In order to properly analyze the set oftests performed in the WMAP maps, we have introducedthe notion of a global anomaly statistic to estimate therelevance of the low-multipoles tests of non-Gaussianityfor each map. A cautionary note is sounded by the globalanomaly statistic: this criterium shows that the CMB mapswe studied have rather low statistics, but they still liewithin the �10% of random maps with lowest globalanomaly statistics, meaning that 90% of the Gaussianmock maps have higher global anomaly statistics thanthe CMB maps.

We confirm the well-known alignment between thequadrupole and the octopole, and we found other signifi-cant levels of alignment between the octopole and thehexadecupole. Moreover, we detected that the hexadecu-

pole and the ‘ � 5 multipole have significantly low levelsof self-alignments (see Table I). Taken individually, thesetests show non-Gaussian features at more than 95% C.L.On the other hand, no significant evidence of self-alignment (which is related to the planarity) was foundfor the octopole.

Regarding correlations with the directions of the cosmicdipole, the ecliptic plane or the galactic plane, we haveconfirmed the correlations of the quadrupole and octopolewith the cosmic dipole, as well as the correlation of theoctopole with the ecliptic plane. For all other multipolesand directions we have found no significant alignments.

In conclusion, we have found intriguing evidence ofnonrandom alignments in the multipoles ‘ � 2� 5, whichhave not only survived, but have indeed been strengthenedby the recently released 3-year WMAP data.

ACKNOWLEDGMENTS

L. R. A. would like to thank Joao Barata and for usefulremarks on the properties of projective spaces, andDominik Schwarz for useful comments on the DopplerQuadrupole. We have benefitted from the use of theHEALPix package [55]. We also acknowledge the use ofthe Legacy Archive for Microwave Background DataAnalysis (LAMBDA, [60]), which is supported by theNASA Office of Space Science. L. R. A. received supportfrom CNPq, Grant No. 475376/2004-8. A. B. is supportedby MCT PCI/DTI/7B. I. S. F. is supported by CAPES. T. V.and C. A. W. acknowledge support from CNPq GrantsNos. 305219/2004-9-FA and 307433/2004-8-FA,respectively.

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alignment tests for low cmb multipoles

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